Ex 1.32 Path functions and state functions, 2.4

Structure and Interpretation of Classical Mechanics

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2. On the other hand, the path function $\bar f[q]$ and the path function $\bar \Gamma (\bar f) \circ \Gamma [q]$ are not necessarily the same. Explain.

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4. Write programs to illustrate the behavior.

~~~

(pp Gamma)

(define ((f-bar q) t)
  (f ((Gamma q) t)))
 
(define ((Gamma-bar h-bar) local)
  ((h-bar (osculating-path local)) (time local)))

(define (q t)
  (sin t))

(define (f-bar q)
  (D q))

(define p
  (literal-function 'q))

(show-expression
 ((f-bar p) 't))

$Dq(t)$

(show-expression
 ((osculating-path ((Gamma p) 't_0)) 't))

$q(t_0) + (t - t_0)Dq(t_0)$

(show-expression
 ((Gamma-bar f-bar) ((Gamma p) 't)))

$Dq(t)$

(define ((GammaBar-fBar-o-Gamma f-bar) q)
  (compose (Gamma-bar f-bar) (Gamma q)))

(show-expression
 (((GammaBar-fBar-o-Gamma f-bar) p) 't))

Two paths that have the same local description up to the $n$ -th derivative are said to osculate with order- $n$ contact. For example, a path and the truncated power series representation of the path up to order $n$ have order- $n$ contact; if fewer than $n$ derivatives are needed by a local-tuple function, the path and the truncated power series representation are equivalent.

(define (g-bar q)
  ((expt D 3) q))

(show-expression
 ((Gamma p) 't))

$\left( \begin{array}{c} t \\ q(t) \\ Dq(t) \end{array} \right)$

(show-expression
 ((g-bar p) 't))

$((D^3 q)(t))$

(show-expression
 ((osculating-path ((Gamma p) 't_0)) 't))

$q(t_0) + (t - t_0)Dq(t_0)$

(show-expression
 (((GammaBar-fBar-o-Gamma g-bar) p) 't))

$0$

— Me@2024-10-16 10:34:35 AM

Physics Town

continues

Ex 1.32 Path functions and state functions, 2.4

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